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Stress distributions in nanocomposite sandwich cylinders reinforced by aggregated

carbon nanotube

Reza Shokri-Oojghaz, Rasool Moradi-Dastjerdi, Hassan Mohammadi, Kamran Behdinan

Version Post-print/accepted manuscript

Citation (published version)

Shokri‐Oojghaz, R. , Moradi‐Dastjerdi, R. , Mohammadi, H. and Behdinan, K. (2019), Stress distributions in nanocomposite sandwich cylinders reinforced by aggregated carbon nanotube. Polym. Compos., 40: E1918-E1927. doi:10.1002/pc.25206.

Publisher’s Statement This is the peer reviewed version of the following article:

Shokri‐Oojghaz, R. , Moradi‐Dastjerdi, R. , Mohammadi, H. and Behdinan, K. (2019), Stress distributions in nanocomposite sandwich cylinders reinforced by aggregated carbon nanotube. Polym. Compos., 40: E1918-E1927. doi:10.1002/pc.25206., which has been published in final form at https://onlinelibrary.wiley.com/doi/epdf/10.1002/pc.25206. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

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For Peer ReviewStress distributions in nanocomposite sandwich cylinders

reinforced by aggregated carbon nanotube

Journal: Polymer Composites

Manuscript ID PC-18-1704.R1

Wiley - Manuscript type: Research Article

Date Submitted by the Author: 13-Nov-2018

Complete List of Authors: Shokri-Oojghaz, Reza ; Department of Mechanical Engineering, Islamic Azad University, Lamerd BranchMoradi-Dastjerdi, Rasool; Advanced Research Laboratory for Multifunctional Light Weight Structures, Department of Mechanical & Industrial Engineering, University of TorontoMohammadi, Hassan ; Department of Mechanical Engineering, Islamic Azad University, Lamerd BranchBehdinan, Kamran ; Advanced Research Laboratory for Multifunctional Light Weight Structures , Department of Mechanical & Industrial Engineering, University of Toronto

Keywords: core-shell polymers, functional polymers, nanocomposites

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Stress distributions in nanocomposite sandwich cylinders

reinforced by aggregated carbon nanotube

Reza Shokri-Oojghaz1, Rasool Moradi-Dastjerdi*,2, Hassan Mohammadi1, Kamran Behdinan2

1 Department of Mechanical Engineering, Islamic Azad University, Lamerd Branch, Lamerd, Iran2 Advanced Research Laboratory for Multifunctional Light Weight Structures, Department of

Mechanical & Industrial Engineering, University of Toronto, Toronto, Canada

Abstract

In order to improve the static response of thick hollow cylinders, a sandwich cylinder with

two carbon nanotube (CNT)-reinforced nanocomposite face sheets are proposed in this paper.

Moreover, due to the use of optimum amount of high cost CNTs, the CNT distribution is

suggested to be functionally graded (FG) along the thickness of cylinder. The stress and

deflection profiles of the proposed sandwich cylinders subjected to internal and external

pressures have been investigated using a finite element method (FEM) based on an

axisymmetric model. The significant effect of formation of CNT agglomerations in the

surrounded matrix is considered and the material properties of the resulted nanocomposite are

estimated by Eshelby-Mori-Tanaka approach. Using the developed axisymmetric FEM model,

the effects of CNT aggregation state, volume fraction and distribution as well as geometrical

dimension and loading condition on the stress and deflection distributions of the

nanocomposite sandwich cylinders have been characterized. The extensive simulations have

revealed that instead of adding higher volume fraction of CNT, the selection of suitable

distribution for CNTs can lead to a nanocomposite sandwich cylinder with less deflection.

Keywords: Stress distribution, Sandwich cylinder, Aggregated carbon nanotube, Finite

element method

* Correspondence to: R. Moradi-Dastjerdi; E-mail: rasoul.moradi@iaukhsh.ac.ir

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1. Introduction

In recent decades, the use of sandwich structures has been growing rapidly and has attracted

much attention from scientific research community. These structures usually consist of a thick

low strength core and two thin stiff outer face sheets. The face sheets have been designed to

resist the applied loads and the core layer is employed to stabilize the structure and to withstand

shear loads [1–3]. Due to the high strength-to-weight ratio and corrosive resistances of

sandwich structures, they are widely used in various industrial applications such as aerospace,

automobile, and biomechanics [4–6]. Moreover, the advantages of sandwich structures can be

intensified by the use of CNT in their face sheet(s) because of extraordinary properties of CNTs

such as extremely high elasticity modulus and low weight [7,8]. The aforementioned

exceptional characteristics of CNT make them a great candidate for reinforcing different

polymer based nanocomposites [7,8]. Therefore, most researchers in this area have focused on

the mechanical properties and analyses of carbon nanotube reinforced composites (CNTRCs)

[9–14]. It has been demonstrated that adding a small amount of CNTs in a polymeric matrix

can considerably improve the mechanical properties of the resulted nanocomposite [15–20].

Also, some research works have presented the successful use of piezoelectric fiber in the smart

damping of CNTRC structures [21–24].

Functionally graded materials (FGMs) are a new class of composite materials with gradient

volume fraction of the components. The use of FGM concept is strongly recommended to

optimize the distribution and volume fraction of CNTs in the matrix. The nanocomposites

reinforced by CNTs with grading distribution are called functionally graded carbon nanotube

reinforced composites (FG-CNTRC) and many researchers have focused on the mechanical

responses of these nanocomposite structures. The earliest work on FG-CNTRC was reported

by Shen [25]. He presented nonlinear bending behavior of nanocomposite plates reinforced by

FG distribution of straight CNTs and proved that using the FGM concept can improve the

reinforcing behavior of CNTs. Sobhani Aragh and Hedayati [26] investigated free vibrational

behavior of FG-CNTRC cylindrical shells reinforced by FG distribution of straight randomly

oriented CNT using a two-dimensional generalized Differential Quadrature method (DQM).

Pourasghar and Kamarian [27] investigated free vibration analysis of cylindrical panels

reinforced by multi-walled carbon nanotubes (MWCNT). They employed modified Halpin-

Tasi approach to estimate material properties of the proposed nanocomposite. Liew et al. [28]

used a mesh-free method and conducted postbuckling analysis of FG-CNTRC cylindrical

panels. They also used the arc-length and Newton-Raphson methods for the postbuckling path.

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Shen and Xiang [29] presented geometrically nonlinear free vibration analysis of FG-CNTRC

cylindrical panels resting on elastic foundations. They used higher-order shear deformation

theory (HSDT) and reported natural frequencies of the panels at different environment

temperatures. Moradi-Dastjerdi et al. [30] developed a refined shear deformation plate theory

to evaluate the natural frequencies of sandwich plates with nanocomposite face sheets

reinforced by FG distribution of randomly oriented straight CNT. Zafarmand and

Kadkhodayan [31] utilized axisymmetric theory of elasticity to study nonlinear static behavior

of FG-CNT reinforced rotating thick disks. Alibeigloo [32] presented thermoelastic stress

distributions in FG-CNTRC cylindrical panels using 3D theory of elasticity. Tahouneh and

Naei [33] employed 3D elasticity solution and presented free vibration analysis of

nanocomposite curved panels and sandwich panels reinforced by FG randomly oriented CNTs

resting on elastic foundation. Safari et al. [34] developed an axisymmetric FE model to study

vibration and damping behavior of hallow FG-CNTRC cylinders filled by magneto-rheological

(MR) oil. Also, a mesh-free method based on MLS shape functions and transformation method

was utilized for static, free vibration and dynamic behavior axisymmetric cylinders reinforced

by FG distribution of wavy CNTs [35–39].

One of the most important concerns in the use of CNTs to reinforce a polymeric matrix is

the formation of CNT agglomerations, especially in high values of CNTs volume fractions

[40]. The main reasons of formation of CNT agglomeration are CNT’s low bending rigidity

and high aspect ratio (usually >1000). It already was proved that formation of CNT

agglomeration sharply decreases the efficiency of CNT reinforcement [41,42]. Pourasghar et

al. [43] utilized Eshelby–Mori–Tanaka approach and 3D theory of elasticity to examine the

effects of CNT orientation and aggregation on the natural frequencies of FG-CNTRC cylinders.

The effect of formation of CNT agglomeration on the sandwich buckling and vibration

behaviors of sandwich plates with two face sheets made of FG-CNTRC were investigated [44–

49]. Sobhaniaragh et al. [50] examined the effect of CNT agglomeration formation on the stress

distribution of ceramic based FG-CNTRC cylindrical shells subjected to thermo-mechanical

loads. They also used based on HSDT to investigate buckling analysis of cylindrical shells

reinforced by aggregated CNT and stiffened by rings and stringers [51]. Tornabene et al. [52–

54] implemented GDQM incorporated with different types of shear deformation theories to

analyze the static and vibration behaviors of nanocomposite shells and plates reinforced by

agglomerated CNTs. Banić et al. [55] utilized Carrera unified formulation (CUF), HSDT and

GDQM to investigate the effect of Winkler-Pasternak elastic foundation on the fundamental

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frequency of FG-CNTRC plates and shells. Using a multiscale approach, the free vibration

behavior of laminated three‐phase nanocomposite shells and plates reinforced with

agglomeration of CNTs and fibers were presented by Tornabene et al. [56]. Moradi-Dastjerdi

et al. [57] also investigated the effect of CNT orientation and aggregation on the static behavior

axisymmetric cylinders under internal pressures.

In this paper, to optimize the utilized volume of CNTs and to decrease the deflection of thick

cylinders subjected to internal and/or external pressure, two solutions are proposed: (i) using

sandwich cylinders with two CNT-reinforced nanocomposite face sheets (ii) the use of FG

concept for the distribution of CNTs in face sheets. As mentioned before, the formation of CNT

agglomerations is one the most important concerns for CNT-reinforced nanocomposites which

was mostly ignored for simplifications. So, we used a modified Eshelby–Mori–Tanaka

approach which is proposed by Shi et al. [58] to estimate the elastic properties of the

nanocomposites reinforced with CNT agglomerations. Moreover, an axisymmetric FEM which

is applicable for thick cylinders was developed to present high accurate results for the proposed

nanocomposite sandwich cylinders. In our simulations of sandwich cylinders, the effect of the

formation of CNT agglomerations on the stress and deflection distributions were investigated

using the developed FE model. Furthermore, the effects of CNT volume fraction and

distribution, geometrical dimensions and loading condition on the stress and deflection

distributions of the nanocomposite sandwich cylinders have been characterized.

2. Material properties in FG-CNTRC face sheets

In this paper, a polymeric sandwich cylinder with two symmetric nanocomposite face sheets

subjected to internal and/or external pressure has been considered as depicted in Fig. 1. In the

sandwich cylinder, ri, ro, h, H and L represent the inner radius, the outer radius, the face sheet

thickness, the cylinder thickness and the cylinder length, respectively. The face sheets are

assumed to be made of a mixture of single walled carbon nanotube (SWCNT) and an isotropic

polymer. Moreover, the formation of CNT clusters in the nanocomposite layers are taken into

account. Also, the distribution of CNTs along the thickness of cylinders are assumed either

uniformly distributed (UD) or functionally graded (FG) in the radial direction. The CNT

volume fraction, fr, are varied along the sandwich cylinder thickness as illustrated in Fig. 2 and

according to [59]:

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max

max

(1 )

0

(1 )

nir r i i

r i o

nor r o o

rrf f r r r hh h

f r h r r hrrf f r h r r

h h

(1)

where n and are volume fraction exponent and maximum value of CNT volume fraction 𝑓𝑟𝑚𝑎𝑥

(which is occurred on the external faces), respectively.

In order to consider the effect of the randomly orientation of CNTs, Shi et al. [58] defined

an average value for all orientations using 3D transformation from local CNT coordinates to

the global coordinates. Then, they obtained the elastic properties of the resulted nanocomposite

from the average strain of a representative volume element (RVE). Moreover, for the effect of

CNT cluster formation on the material properties of randomly oriented CNT-reinforced

composite, Shi et al. [58] defined a two extra parameters in Eshelby–Mori–Tanaka approach.

They assumed that the total volume of nanotubes inside the RVE, Vr, are uniformly distributed

into some clusters and the rest of them are located in the matrix and outside the clusters clusterrV

. Therefore, the aggregation states can be defined by the following two parameters:mrV

/ , / 0 , 1clustercluster r rV V V V (2)

where V and Vcluster are the volumes of RVE and clusters in the RVE, respectively. Hence,

shows the volume fraction of clusters inside the RVE, and η describes the amount of CNTs

located outside the clusters. It should be mentioned that indicates uniform distribution of 1

CNTs inside the composite without cluster and decreasing describe a nanocomposite with

severe agglomeration degree. Moreover, indicates all the nanotubes are located in the 1

clusters. The case means that the volume fraction of CNTs inside the clusters is as same

as that of CNTs outside the clusters (fully-dispersed).

Thus, it is assumed that the CNT-reinforced composite is a system consisting of spherical

CNT clusters embedded in a matrix. In this approach, first, the mechanical properties of inside

and outside of clusters are estimates. The estimation is followed by calculation of the overall

properties of the nanocomposite RVE. For the clusters, Prylutskyy et al. [60] calculated the

effective bulk modulus, Kin, and shear modulus, Gin, as given:

,

3

3r r m r

in mr r r

f KK K

f f

3

2 3r r m r

in mr m r

f GG G

G

(3)

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where:

33

m m r rr

m r

K G k lK k

(4)

2 (3 ) (3 7 )4 2 41

5 3 (3 ) (3 7 )m m m m m mm r r m

rm r m r m m m r m m

G K G G K GG k l GG k G p G K G m K G

(5)

(2 ) (3 2 )1 23

r r m m rr r r

m r

k l K G ln lG k

(6)

8 8 (3 4 ) 2( )(2 )1 2 ( )5 3 3 ( ) (7 ) 3( )

m r r m m m r r m rr r r

m r m r m m r m m r

G p m G K G k l G ln lG p K m G G m G G k

(7)

kr, lr, mr, nr, and pr are the Hill’s elastic moduli for the reinforcing phase (transversely isotropic

CNTs) which are reported in [58]. Also, those effective parameters for the outside of the

clusters can be estimated as [58]:

,

(1 ) 3

3 1 (1 ) (1 )r r m r

out mr r r

f KK K

f f

(1 ) 22 1 (1 ) (1 )

r r m rout m

r r r

f GG G

f f

(8)

fm (=1-fr) is the matrix volume fractions. Finally, the effective bulk modulus K and the effective

shear modulus G of the composite are derived from the Mori-Tanaka method as follows [58]:

,

11

1 (1 ) 1

in

outout

in

out

KK

K KKK

11

1 (1 ) 1

in

outout

in

out

GKG

G GGG

(9)

with

, , 3 2

2(3 )out out

outout out

K GK G

1

3(1 )out

out

2(4 5 )15(1 )

out

out

(10)

The effective Young’s modulus E and Poisson’s ratio of the composite in the terms of K

and G are given by:

, 93

KGEK G

3 26 2

K GK G

(11)

3. Governing equations

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The weak form of equilibrium equation for the sandwich cylinder subjected to mechanical

loads can be expressed [57]:

0)( dsvd uFεσ (12)

where , , and are the vectors of stress, strain, displacement and surface traction, σ ε u F

respectively. is a part of boundary of domain on which traction is applied. For Faxisymmetric problems stress and strain vectors are as follows:

, , , , , , ,T Tr z rz r z rz σ ε (13)

The stress vector is expressed in terms of strain vector and stiffness matrix using Hook's Dlaw:

Dεσ (14)

As mentioned before, the resulted nanocomposites are isotropic materials. Therefore, D is

given as follows:

1 01 0

1 0(1 )(1 2 )(1 2 )0 0 0

2

E

D (15)

For FG-CNTRC sandwich cylinders, E and are (radial) location-dependent. Moreover, in axisymmetric problems, the strain vector components are defined in terms of the displacement

components as follows:

, , ,r r z r zr z rz

u u u u ur r z z r

(16)

4. Finite element formulation

Using axisymmetric finite element shape functions N, the displacement vector in the energy

function can be approximated as follows:

NuU Tzr uu ],[ (17)

where nodal values vector and shape function matrix , are described as:u N

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Tnznrzr uuuu )(,)..(,.........)(,)( 11u (18)

n

nNNN

NNN0....00

0....00

21

21N (19)

n is total number of nodes. The strain vector, Eq. (16), can be rewritten by employing Eq. (17)

in FE form as [34]:

ε Bu (20)

where

r

N

z

N...

z

N0...

0r

N...

0r

N...

..r

Nz

Nr

Nz

N

..z

N0

zN

0

..0r

N0

rN

..0r

N0

rN

pp

p

p

p

2211

21

21

21

B (21)

p is equal to the node numbers of each elements. Finally, by substitution of Eq. (20), Eq. (17)

and Eq. (14) in Eq. (12) leads to:

ku F (22)

in which, stiffness matrix k and force vector F of the nanocomposite sandwich cylinder are

represented as follows:

,T Tdv f dA

k B DB F N (23)

Solving the system Eq. (22) gives the displacement field. Then, using Eqs. (20) and (14), the

strain and stress distributions could be derived.

5. Results and discussions

In this section, at first, the accuracy of the utilized FE method on the static behavior of

cylinders are examined. Then, the distributions of stress and deflection in sandwich cylinders

with nanocomposite face sheets subjected to internal and external pressures are conducted. The

nanocomposite face sheets are made from (10,10) SWCNT agglomerations embedded in

PMMA. PMMA is an isotropic polymer material with , and 2.5mE GPa 31150 /m Kg m

[25]. Moreover, the considered SWCNT is a transversely isotropic nano-filler with0.34m

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, , , and 1 5.6466CNE TPa 2 7.0800CNE TPa 12 1.9445CNG TPa 31400 /CN Kg m 175.012 CN

[25].

5.1. Validation of the model

The exact distributions of radial stress, axial stress, and radial displacement in a long length

(plane strain) isotropic thick cylinder subjected to internal and external pressures are reported

by Sadd [61] as follows:

, , 2rA Br

2A Br

21 (1 2 )r

AU r BE r

(24)

where

2 2 2 2

2 2 2 2( ) ,i o o i i i o o

o i o i

r r P P r P r PA Br r r r

(25)

Figure 3 compares the obtained stresses and displacement distributions for a cylinder made

of PMMA with inner radius of , and outer radius of , subjected to internal 0.1ir m 0.2or m

pressure of . An excellent agreement between the FEM results and exact solution 10iP MPa

can be seen which proves the high accuracy of the developed FEM. Moreover, the accuracy of

the developed finite element method was also verified for stress distribution in FGM cylinders

by comparison with analytical and numerical methods in [37,57].

5.2. Static analysis of long length sandwich cylinders with FG-CNTRC face sheets

In order to investigate the effects of aggregation state (μ and η), first, we considered

sandwich cylinders with two CNTRC face sheets which distribution of CNTs along the face

sheets thicknesses are assumed to be linear, n=1, and subjected to internal pressure of

. The sandwich cylinders have outer radius of, , radii ratio of, , 10iP MPa 0.2or m / 0.5i or r

ratio of face sheets thickness to total thickness of, , and maximum CNT volume / 0.2h H

fraction of, . Figure 4 extensively shows the effects of μ and η on the internal max 0.05rf

deflection of cylinders. It can be seen that increasing the CNT agglomeration volume (μ) or

decreasing CNT concentration inside the agglomerations (decreasing η from unit) leads to

decrease in the deflection values of the sandwich cylinders. Because by increasing μ or

decreasing η, the distribution of CNT in the surrounded polymeric matrix is improved which

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results in to better enhancement in the mechanical properties of sandwich cylinders. Moreover,

Figure 4 illustrates that sandwich cylinders with aggregation state of μ=η which determines

fully distribution of CNTs inside the matrix, has minimum deflection.

Table 1 represents the internal deflections of sandwich cylinders with , 0.2or m

, and η=1 to investigate the effects of cylinder thickness, face sheets 10iP MPamax 0.05rf

thickness, CNT distribution and agglomeration states. It can be seen that due to decrease in the

total volume of CNTs embedded inside the polymer, the increase of exponent of CNT

distribution n increases the deflection of sandwich cylinder. Moreover, as expected, the

increase of face sheet thicknesses or the decrease of radii ratio decreases the deflection.

Furthermore, Figure 5 illustrates the distributions of radial displacement, hoop stress and axial

stress along the thickness of the same sandwich cylinders for μ =0.1 and μ =0.4. It is disclosed

that the increase of agglomeration volume, due to improvement of CNT distribution and then

the mechanical properties of resulted nanocomposite, decreases the values of deflections and

increases the values of hoop stresses along the thickness, while, it has not a considerable effect

on the radial stress distribution. The values of radial stresses vary from internal pressure,

, at inner radius to zero at outer radius. It is well established that radial displacement 10iP MPa

and stress should be continuous and smooth. Otherwise, it shows that separation between the

layers of sandwich cylinder (delamination) has happened.

CNT volume fraction and loading conditions are two other parameters which have a

significant effect on the static behavior of CNT-reinforced nanocomposite sandwich cylinders.

Table 2 presents the effect of two above mentioned parameters on the internal radial deflection

of long sandwich cylinders with , , , μ =0.2 and η=1 for 0.2or m / 0.5i or r / 0.2h H

different values of CNT volume fraction exponents. It is observed that when , the max 0rf

isotropic cylinder without nano-filler, as expected, the exponent of CNT volume fraction has

no effects on the cylinder deflection. However, using 2% of CNT leads to a significant

reduction in the radial deflection and also the decrease of exponent of CNT volume fraction

improves the mentioned positive effect. It is worth noting that increasing CNT volume from

2% to 5% does not considerably reduce the deflection of the cylinder, while, CNT exponent n

remarkably decreases it. It means that instead of using more values of high cost CNT volume,

the selection of a proper profile for CNT distributions can offer more reduction in the deflection

of the nanocomposite sandwich cylinders. It is also observed that the increase of internal

pressure regularly increases the values of deflections. However, adding external pressure

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remarkably decreases the deflection values. Moreover, Figure 6 shows the variations of

deflection and stresses along the thickness of the same sandwich cylinders when n=1,

, and for and . It can be seen that the use of CNTs in the 10iP MPa 5oP MPa 10oP MPa

cylinders considerably reduces the values of radial deflections because it makes a

nanocomposite sandwich cylinder from an isotropic cylinder. In sandwich cylinders, the values

of hoop stresses in nanocomposite face sheets and polymeric core are more and less than the

corresponding isotropic cylinders, respectively. However, the gradients of hoop stress in

sandwich cylinders are much more than corresponding isotropic cylinders. This phenomenon

is also observed for radial stress distribution when the values of internal and external pressure

are equals.

5.3. Static analysis of finite length sandwich cylinders with FG-CNTRC face sheets

As mentioned before, all of the previous sandwich cylinders were in plane strain condition.

In order to examine the effect of cylinder length on the static response, clamped-clamped finite

length nanocomposite sandwich cylinders subjected to internal pressure loads are assumed.

Figure 7 shows the deflection and stress distributions along the radius of nanocomposite

sandwich cylinders with , , , n=1, , μ =0.2 and 0.2or m / 0.5i or r / 0.2h H max 0.05rf

η=1 for the different ratio of length to outer radius L/ro. It can be seen that by increasing the

ratio of L/ro, the values of deflection and stresses are increased. However, for the values of

higher than L/ro=3, this increase is disappeared and the results are converged to the results of

nanocomposite sandwich cylinders with plane strain condition. The reason is that by increasing

the length of cylinders, the effect of essential boundary conditions on the static response of the

structures is decreased.

6. Conclusions

In this paper, static response of sandwich cylinders with two nanocomposite face sheets

under internal and external pressures were investigated using FEM based on an axisymmetric

model. The face sheets of the sandwich cylinders were made of a mixture of FG distributed

CNT and polymer nanocomposite. The effect of forming of CNT agglomeration was

considered and the material properties were estimated using Eshelby-Mori-Tanaka approach.

The effects of CNT aggregation state, volume fraction and distribution as well as geometrical

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dimension and loading condition on the static response of the nanocomposite sandwich

cylinders were investigated and the following results were obtained:

Selection of suitable distribution for CNTs offers a sandwich cylinder with less

deflection compared with sandwich cylinders with higher values of CNT volume;

Increase of agglomeration volume or decrease of CNT concentration inside the

agglomerations improve CNT distribution and accordingly decrease the deflection

values of sandwich cylinders;

The hoop stresses in cylinder increases as the CNT agglomerate size increases;

In most cases, radial stress along the thickness smoothly changes from the value of

internal pressure at the inner radius to the value of external pressure at outer radius;

Use of CNTs in the cylinders considerably decreases the values of deflection, however,

increases the gradients of hoop stress in sandwich cylinders;

Increase of the length of cylinder increases the values of deflection and stresses.

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[24] Suresh Kumar R, Kundalwal SI, Ray MC. Control of large amplitude vibrations of doubly curved sandwich shells composed of fuzzy fiber reinforced composite facings. Aerosp Sci Technol 2017;70:10–28. doi:10.1016/j.ast.2017.07.027.

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Graded Wavy Carbon Nanotube-Reinforced Cylinders. Polym Compos 2018;39:E826–E834. doi:10.1002/pc.24278.

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doi:10.1016/j.compositesb.2015.11.016.

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Figure Captions

Fig. 1 Schematic of the sandwich cylinder with nanocomposite face sheets reinforced by

aggregated CNT

Fig. 2 Functionally graded distribution of CNTs along the thickness of sandwich cylinders

Fig. 3 Comparison between the obtained distributions of (a) radial displacement (b) hoop

stress (c) radial stress from FEM and exact solution for a PMMA cylinder with , 0.1ir m

and 0.2or m 10iP MPa

Fig. 4 Internal deflection versus μ for sandwich cylinders with n=1, , , 0.2or m / 0.5i or r

, and in different η/ 0.2h H max 0.05rf 10iP MPa

Fig. 5 Distributions of (a) radial displacement (b) hoop stress (c) radial stress along the

thickness of the sandwich cylinders with , , and 0.2or mmax 0.05rf 10iP MPa

Fig. 6 Distributions of (a) radial displacement (b) hoop stress (c) radial stress along the

thickness of the sandwich cylinders with , , , n=1, μ 0.2or m / 0.5i or r / 0.2h H

=0.2, η=1and 10iP MPa

Fig. 7 Distributions of (a) radial displacement (b) hoop stress (c) radial stress along the

thickness of finite length sandwich cylinders with , , , 0.2or m / 0.5i or r / 0.2h H

n=1, , μ =0.2, η=1 and max 0.05rf 10iP MPa

Table Captions

Table 1 Internal deflections of the sandwich cylinders with , , 0.2or m 10iP MPa

and η=1max 0.05rf

Table 2 Internal deflections of the sandwich cylinders with , , 0.2or m / 0.5i or r

, μ =0.2 and η=1/ 0.2h H

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Figures

Fig. 1 Schematic of the sandwich cylinder with nanocomposite face sheets reinforced by

aggregated CNT

Fig. 2 Functionally graded distribution of CNTs along the thickness of sandwich cylinders

FG-C

NT

RC

FG-C

NT

RC

Isot

ropi

c C

ore

ri

ro

H

hhL

r

z

CNT ClusterCNTMatrix

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(a)

(b)

(c)

Fig. 3 Comparison between the obtained distributions of (a) radial displacement (b) hoop

stress (c) radial stress from FEM and exact solution for a PMMA cylinder with , 0.1ir m

and 0.2or m 10iP MPa

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Fig. 4 Internal deflection versus μ for sandwich cylinders with

n=1, , , , and in different η0.2or m / 0.5i or r / 0.2h H max 0.05rf 10iP MPa

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(a)

(b)

(c)

Fig. 5 Distributions of (a) radial displacement (b) hoop stress (c) radial stress along the

thickness of the sandwich cylinders with , , and 0.2or mmax 0.05rf 10iP MPa

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(a)

(b)

(c)

Fig. 6 Distributions of (a) radial displacement (b) hoop stress (c) radial stress along the

thickness of the sandwich cylinders with , , , n=1, μ =0.2, 0.2or m / 0.5i or r / 0.2h H

η=1and 10iP MPa

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(a)

(b)

(c)Fig. 7 Distributions of (a) radial displacement (b) hoop stress (c) radial stress along the

thickness of finite length sandwich cylinders with , , , n=1, 0.2or m / 0.5i or r / 0.2h H

, μ =0.2, η=1 and max 0.05rf 10iP MPa

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Tables

Table 1 Internal deflections of the sandwich cylinders with

, , and η=10.2or m 10iP MPamax 0.05rf

/i or r /h H μ n=0.1 n=1 n=100.5 0.1 0.1 3.6406 3.6468 3.7739

0.2 3.4132 3.4364 3.70200.4 2.9177 3.0022 3.5580

0.2 0.1 3.4760 3.4871 3.69840.2 3.1091 3.1457 3.56610.4 2.4101 2.5182 3.3120

0.7 0.1 0.1 8.1143 8.1265 8.36380.2 7.6876 7.7342 8.23780.4 6.7120 6.8903 7.9780

0.2 0.1 7.7694 7.7935 8.22510.2 7.0355 7.1169 7.98630.4 5.5741 5.8248 7.5131

Table 2 Internal deflections of the sandwich cylinders

with , , , μ =0.2 and η=10.2or m / 0.5i or r / 0.2h H

( )iP MPa ( )oP MPa max (%)rf n=0.1 n=1 n=10

5 0 0 1.9296 1.9296 1.92962 1.5670 1.5999 1.81155 1.5546 1.5729 1.7831

10 0 0 3.8592 3.8592 3.85922 3.1339 3.1998 3.62315 3.1092 3.1457 3.5661

10 5 0 1.5008 1.5008 1.50082 1.2007 1.2260 1.40035 1.1911 1.2050 1.3768

10 10 0 -0.8576 -0.8576 -0.85762 -0.7326 -0.7477 -0.82255 -0.7269 -0.7357 -0.8126

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Table 1 Internal deflections of the sandwich cylinders with

0.2or m= , 10iP MPa= , max 0.05rf = and η=1

/i or r /h H µ n=0.1 n=1 n=10

0.5 0.1 0.1 3.6406 3.6468 3.7739

0.2 3.4132 3.4364 3.7020

0.4 2.9177 3.0022 3.5580

0.2 0.1 3.4760 3.4871 3.6984

0.2 3.1091 3.1457 3.5661

0.4 2.4101 2.5182 3.3120

0.7 0.1 0.1 8.1143 8.1265 8.3638

0.2 7.6876 7.7342 8.2378

0.4 6.7120 6.8903 7.9780

0.2 0.1 7.7694 7.7935 8.2251

0.2 7.0355 7.1169 7.9863

0.4 5.5741 5.8248 7.5131

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Table 2 Internal deflections of the sandwich cylinders

with 0.2or m= , / 0.5i or r = , / 0.2h H = , µ =0.2 and η=1

( )iP MPa ( )oP MPa max (%)rf n=0.1 n=1 n=10

5 0 0 1.9296 1.9296 1.9296

2 1.5670 1.5999 1.8115

5 1.5546 1.5729 1.7831

10 0 0 3.8592 3.8592 3.8592

2 3.1339 3.1998 3.6231

5 3.1092 3.1457 3.5661

10 5 0 1.5008 1.5008 1.5008

2 1.2007 1.2260 1.4003

5 1.1911 1.2050 1.3768

10 10 0 -0.8576 -0.8576 -0.8576

2 -0.7326 -0.7477 -0.8225

5 -0.7269 -0.7357 -0.8126

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Fig. 1 Schematic of the sandwich cylinder with nanocomposite face sheets reinforced by aggregated CNT

122x59mm (96 x 96 DPI)

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Fig. 2 Functionally graded distribution of CNTs along the thickness of sandwich cylinders

133x84mm (96 x 96 DPI)

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Fig. 3 Comparison between the obtained distributions of (a) radial displacement (b) hoop stress (c) radial stress from FEM and exact solution for a PMMA cylinder with , and

69x156mm (96 x 96 DPI)

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Fig. 4 Internal deflection versus μ for sandwich cylinders with n=1, , , , and in different η

133x84mm (96 x 96 DPI)

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Fig. 5 Distributions of (a) radial displacement (b) hoop stress (c) radial stress along the thickness of the sandwich cylinders with , , and

131x162mm (96 x 96 DPI)

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Fig. 6 Distributions of (a) radial displacement (b) hoop stress (c) radial stress along the thickness of the sandwich cylinders with , , , n=1, μ =0.2, η=1and

132x162mm (96 x 96 DPI)

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Fig. 7 Distributions of (a) radial displacement (b) hoop stress (c) radial stress along the thickness of finite length sandwich cylinders with , , , n=1, , μ =0.2, η=1 and

67x142mm (96 x 96 DPI)

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